I noticed that each of the class achievements require you to get to level 60 with two separate characters of the same class, but you only get 9 character slots. Does that mean it's not possible to get every single achievement in the game?
The specific achievement you mention is obtainable as there are actually 10 character slots. You can also get this achievement by creating, leveling, then deleting characters (or selling them, after Blizzard allows characters to be sold on the AH).
Regarding the rest, it is indeed possible to get all achievements; none of them should be mutually exclusive.
In fact, here is a thread where a number of people discuss their attempts to get the "world first" for obtaining all achievements.
The exception is that, as @aolszowka mentions, some Feats of Strength are no longer obtainable. However, these are usually excluded from "all achievements," and in fact this is one of the reason Blizzard invented feats of strength.
When you say every achievement are you speaking to every achievement that has some type of point value? Or just every achievement period? If so you might have already been out of luck. The feats of strength achievements for example require that you:
- Were a part of the Diablo 3 Closed Beta
- Participated in the Web Promotion
- Purchased the Collectors Edition
Two of the above are now impossible to achieve, and the third might as well be with the current asking price on Amazon/eBay.
To answer your direct question with regards to the Class Achievements, I guess Blizzard would assume that you would roll a Hardcore Character at some point, and inevitably die as well. My understanding is that this opens up another character slot to allow you to continue to try for the achievement.
The achievements the OP asks about are all possible, but there are still a couple of achievements that are impossible due to a bug which has not yet been fixed (as of 9 August 2012). In Act 3, the lorebook "Battlefield Reports" never drops, which means it's impossible to get the "Espionage" achievement, as well as one or two other achievements which depend on that one.